Theta operators, Goss polynomials, and v-adic modular forms

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

arXiv:1605.04235v1 [math.NT] 13 May 2016

MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG In honor of David Goss Abstract. We investigate hyperderivatives of Drinfeld modular forms and determine formulas for these derivatives in terms of Goss polynomials for the kernel of the Carlitz exponential. As a consequence we prove that v-adic modular forms in the sense of Serre, as defined by Goss and Vincent, are preserved under hyperdifferentiation. Moreover, upon multiplication by a Carlitz factorial, hyperdifferentiation preserves v-integrality.

1. Introduction In [21], Serre introduced p-adic modular forms for a fixed prime p, as p-adic limits of Fourier expansions of holomorphic modular forms on SL2 (Z) with rational coefficients. He established fundamental results about families of p-adic modular forms by developing the theories of differential operators and Hecke operators acting on p-adic spaces of modular forms, and in particular he showed that the weight 2 Eisenstein series E2 is also p-adic. If 1 d we let ϑ := 2πi be Ramanujan’s theta operator acting on holomorphic complex forms, dz then letting q(z) = e2πiz , we have (1.1)

ϑ=q

d , dq

ϑ(qn ) = nqn .

Although ϑ does not preserve spaces of complex modular forms, Serre proved the induced operation ϑ : Q ⊗ Zp [[q]] → Q ⊗ Zp [[q]] does take p-adic modular forms to p-adic modular forms and preserves p-integrality. In the present paper we investigate differential operators on spaces of v-adic modular forms, where v is a finite place corresponding to a prime ideal of the polynomial ring A = Fq [θ], for Fq a field with q elements. Drinfeld modular forms were first studied by Goss [10], [11], [12], as rigid analytic functions, f : Ω → C∞ , on the Drinfeld upper half space Ω that transform with respect to the group Γ = GL2 (A) (see §4 for precise definitions). Here if we take K = Fq (θ), then Ω is defined to be C∞ \ K∞ , where K∞ = Fq ((1/θ)) is the completion of K at its infinite place and C∞ is the completion of an algebraic closure of K∞ . Goss showed that Drinfeld modular forms have expansions in terms of the uniformizing parameter u(z) := 1/eC (e πz) at the infinite Date: May 13, 2016. 2010 Mathematics Subject Classification. Primary 11F52; Secondary 11F33, 11G09. Key words and phrases. Drinfeld modular forms, Goss polynomials, v-adic modular forms, hyperderivatives, false Eisenstein series. This project was partially supported by NSF Grant DMS-1501362. 1

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MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

cusp of Ω, where eC (z) is the exponential function of the Carlitz module and π e is the Carlitz period. Each such form f is uniquely determined by its u-expansion, ∞ X f= cn un ∈ C∞ [[u]]. n=0

If k ≡ 0 (mod q − 1), then the weight k Eisenstein series of Goss [11], has a u-expansion due to Gekeler [7, (6.3)] of the form X ζC (k) ζC (k) Gk (u(az)), ∈ K, Ek = − k − π e π ek a∈A, a monic

where ζC (k) is a Carlitz zeta value, Gk (u) is a Goss polynomial of degree k for the lattice ΛC = Ae π (see §3–4 and (4.2)), and u(az) can be shown to be represented as a power series in u (see §4). Gekeler and Goss also show that spaces of forms for Γ are generated by forms with u-expansions with coefficients in A. Using this as a starting point, Goss [14] and Vincent [24] defined v-adic modular forms in the sense of Serre by taking v-adic limits of u-expansions and thus defining v-adic forms as power series in K ⊗A Av [[u]] (see §5). Goss [14] constructed a family of v-adic forms based on forms with A-expansions due to Petrov [19] (see Theorem 6.3), and Vincent [24] showed that forms for the group Γ0 (v) ⊆ GL2 (A) with v-integral u-expansions are also v-adic modular forms. It is natural to ask how Drinfeld modular forms and v-adic forms behave under differentiation, and since we are in positive characteristic it is favorable to use hyperdifferential dr d d operators ∂zr , rather than straight iteration dz r = dz ◦ · · · ◦ dz (see §2 for definitions). d = − πe1 ∂z1 , then we have the action on Gekeler [7, §8] showed that if we define Θ := − πe1 dz u-expansions determined by the equality d (1.2) Θ = u2 = u2 ∂u1 . du Now as in the classical case, derivatives of Drinfeld modular forms are not necessarily modular, but Bosser and Pellarin [1], [2], showed that hyperdifferential operators ∂zr preserve spaces of quasi-modular forms, i.e., spaces generated by modular forms and the false Eisenstein series E of Gekeler (see Example 4.7), which itself plays the role of E2 . For r > 0, following Bosser and Pellarin we define the operator Θr by 1 Θr := ∂r . (−e π )r z Uchino and Satoh [22, Lem. 3.6] proved that Θr takes functions with u-expansions to functions with u-expansions, and Bosser and Pellarin [1, Lem. 3.5] determined formulas for the expansion of Θr (un ). If we consider the r-th iterate of the classical ϑ-operator, ϑ◦r = ϑ ◦ · · · ◦ ϑ, then clearly by (1.1), ϑ◦r (q n ) = nr q n . If we iterate Θ, taking Θ◦r = Θ ◦ · · · ◦ Θ, then by (1.2) we find   n + r − 1 n+r ◦r n u , Θ (u ) = r! r

which vanishes identically when r > p. On the other hand, the factor of r! is not the only discrepancy in comparing Θr and Θ◦r , and in fact we prove two formulas in Corollary 4.10

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

3

revealing that Θr is intertwined with Goss polynomials for ΛC :
(1.3) Θr (un ) = un ∂un−1 un−2 Gr+1 (u) , ∀ n > 1,  r  X n+j −1 r n (1.4) βr,j un+j , ∀ n > 0, Θ (u ) = j j=0 where βr,j are the coefficients of Gr+1 (u). These formulas arise from general results (Theorem 3.4) on hyperderivatives of Goss polynomials for arbitrary Fq -lattices in C∞ , which is the primary workhorse of this paper, and they induce formulas for hyperderivatives of u-expansions of Drinfeld modular forms (Corollary 4.12). It is important to note that (1.4) is close to a formula of Bosser and Pellarin [1, Eq. (28)], although the connections with coefficients of Goss polynomials appears to be new and the approaches are somewhat different. Goss [14] defines the weight space of v-adic modular forms to be S = Z/(q d − 1)Z × Zp , where d is the degree of v, and if we take Mm s ⊆ K ⊗A Av [[u]] to be the space of v-adic forms of weight s ∈ S and type m ∈ Z/(q − 1)Z (see §5), then we prove (Theorem 6.1) that Θr preserves spaces of v-adic modular forms, m+r Θr : M m s → Ms+2r ,

r > 0.

Of particular importance here is proving that the false Eisenstein series E is a v-adic form (Theorem 6.5). Unlike in the classical case, Θr does not preserve v-integrality due to denominators coming from Gr+1 (u), but we show in §7 that this failure can be controlled, namely showing (Theorem 7.4) that m+r Πr Θr : Mm s (Av ) → Ms+2r (Av ),

r > 0,

m where Πr ∈ A is the Carlitz factorial (see §2) and Mm s (Av ) = Ms ∩ Av [[u]].

Acknowledgements. The authors thank David Goss for a number of helpful suggestions on an earlier version of this article. 2. Functions and hyperderivatives Let Fq be the finite field with q elements, q a fixed power of a prime p. Let A := Fq [θ] be a polynomial ring in one variable, and let K := Fq (θ) be its fraction field. We let A+ denote the monic elements of A, Ad+ the monic elements of degree d, and A( 1, we set (2.1)

i

[i] = θq − θ,

Di = [i][i − 1]q · · · [1]q

i−1

,

Li = (−1)i [i][i − 1] · · · [1],

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MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

q and we let D0 = L0 = 1. We have the recursions, Di = [i]Di−1 and Li = −[i]Li−1 , and we recall [13, Prop. 3.1.6] that Y Y (2.2) [i] = f, Di = a, Li = (−1)i · lcm(f ∈ Ai+ ). f ∈A+ , irred. deg(f )|i

a∈Ai+

For m ∈ Z+ , we define the Carlitz factorial Πm as follows. If we write m = 0 6 mi 6 q − 1, then Y (2.3) Πm = Dimi .

P

mi q i with

i

For more information about Πm the reader is directed to Goss [13, §9.1]. For an Fq -algebra L, we let τ : L → L denote the q-th power Frobenius map, and we let L[τ ] denote the ring of twisted polynomials over L, subject to the condition that τ c = cq τ for c ∈ L. We then define as usual the Carlitz module to be the Fq -algebra homomorphism C : A → A[τ ] determined by Cθ = θ + τ. The Carlitz exponential is the Fq -linear power series, (2.4)

eC (z) =

i ∞ X zq

i=0

Di

.

The induced function eC : C∞ → C∞ is both entire and surjective, and for all a ∈ A, eC (az) = Ca (eC (z)). The kernel ΛC of eC (z) is the A-lattice of rank 1 given by ΛC = Ae π , where for a fixed (q − 1)-st root of −θ, ∞  −1 Y
1/(q−1) 1−q i π e = θ(−θ) 1−θ ∈ K∞ (−θ)1/(q−1) i=1

is called the Carlitz period (see [13, §3.2] or [18, §3.1]). Moreover, we have a product expansion   Y′  Y′  z z =z , 1− 1− (2.5) eC (z) = z λ ae π a∈A λ∈Λ C

where the prime indicates omitting the a = 0 term in the product. For more information about the Carlitz module, and Drinfeld modules in general, we refer the reader to [13, Chs. 3–4]. We will say that a function f : Ω → C∞ is holomorphic if it is rigid analytic in the sense of [6]. We set H(Ω) to be the set of holomorphic functions on Ω. We define a holomorphic function u : Ω → C∞ by setting 1 (2.6) u(z) := , eC (e π z) and we note that u(z) is a uniformizing parameter at the infinite cusp of Ω (see [7, §5]), which plays the role of q(z) = e2πiz in the classical case. The function u(z) is A-periodic

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

5

in the sense that u(z + a) = u(z) for all a ∈ A. The imaginary part of an element z ∈ C∞ is set to be |z|i = inf |z − x|∞ , x∈K∞

which measures the distance from z to the real axis K∞ ⊆ C∞ . We will say that an A-periodic holomorphic function f : Ω → C∞ is holomorphic at ∞ if we can write a convergent series, ∞ X cn u(z)n , cn ∈ C∞ , |z|i ≫ 0. f (z) = n=0 P The function f is then determined by the power series f = cn un ∈ C∞ [[u]], and we call this power series the u-expansion of f and the coefficients cn the u-expansion coefficients of f . We set U(Ω) to be the subset of H(Ω) comprising functions on Ω that are Aperiodic and holomorphic at ∞. In other words, U(Ω) consists of functions that have u-expansions. We now define hyperdifferential operators and hyperderivatives (see [5], [15], [22] for more details). For a field F and an independent variable z over F , for j > 0 we define the j-th hyperdifferential operator ∂zj : F [z] → F [z] by setting   n n−j j n z , n > 0, ∂z (z ) = j
where nj ∈ Z is the usual binomial coefficient, and extending F -linearly. (By usual con
vention nj = 0 if 0 6 n < j.) For f ∈ F [z], we call ∂zj (f ) ∈ F [z] its j-th hyperderivative. Hyperderivatives satisfy the product rule,

∂zj (f g)

(2.7)

=

j X

∂zk (f )∂zj−k (g),

f, g ∈ F [z],

k=0

and composition rule, (2.8)

(∂zj

◦

∂zk )(f )

=

(∂zk

◦

∂zj )(f )

  j + k j+k ∂z (f ), = j

f ∈ F [z].

Using the product rule one can extend to ∂zj : F (z) → F (z) in a unique way, and F (z) together with the operators ∂zj form a hyperdifferential system. If F has characteristic 0, dj then ∂zj = j!1 dz Furthermore, j , but in characteristic p this holds only for j 6 p − 1. hyperderivatives satisfy a number of differentiation rules (e.g., product, quotient, power, chain rules), which aid in their description and calculation (see [15, §2.2], [18, §2.3], for a complete list of rules and historical accounts). Moreover, if f ∈ F (z) is regular at c ∈ F , then so is ∂zj (f ) for each j > 0, and it follows that we have a Taylor expansion, ∞ X (2.9) f (z) = ∂zj (f )(c) · (z − c)j ∈ F [[z − c]]. j=0

In this way we can also extend ∂zj uniquely to ∂zj : F ((z − c)) → F ((z − c)). For a holomorphic function f : Ω → C∞ , it was proved by Uchino and Satoh [22, §2] that we can define a holomorphic hyperderivative ∂zj (f ) : Ω → C∞ (taking F = C∞ in the preceding paragraph). That is, ∂zj : H(Ω) → H(Ω).

6

MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

Moreover they prove that the system of operators ∂zj on holomorphic functions inherits the same differentiation rules for hyperderivatives of polynomials and power series. Thus for f ∈ H(Ω) and c ∈ Ω, we have a Taylor expansion, f (z) =

∞ X

∂zj (f )(c) · (z − c)j ∈ C∞ [[z − c]].

j=0

We have the following crucial lemma for our later considerations in §4, where we find new identities for derivatives of functions in U(Ω). Lemma 2.10 (Uchino-Satoh [22, Lem. 3.6]). If f ∈ H(Ω) is A-periodic and holomorphic at ∞, then so is ∂zj (f ) for each j > 0. That is, ∂zj : U(Ω) → U(Ω),

j > 0.

We recall computations involving u(z) and ∂z1 (u(z)) (see [7, §3]). First we see from (2.4) that ∂z1 (eC (z)) = 1, so using (2.5) and taking logarithmic derivatives, 1 ∂ 1 (eC (e π z)) 1X 1 1 = · z = . (2.11) u(z) = eC (e π z) π e eC (e π z) π e a∈A z + a

Furthermore, (2.12)

∂z1 (u(z))

=

∂z1



1 eC (e π z)



=

−∂z1 (eC (e π z)) = −e π u(z)2 . 2 eC (e π z)

Thus, ∂z1 (u) = −e π u2 ∈ U(Ω). In §4 we generalize this formula and calculate ∂zr (un ) for r, n > 0. We conclude this section by discussing some properties of hyperderivatives Ps particular to positive characteristic. Suppose char(F ) = p > 0. If we write j = i=0 bi pi , with 0 6 bi 6 p − 1 and bs 6= 0, then (see [15, Thm. 3.1]) (2.13)

s

∂zj = ∂zb0 ◦ ∂zb1 p ◦ · · · ◦ ∂zbs p ,

which follows from the composition law and Lucas’s theorem (e.g., see [1, Eq. (14)]). We note that for 0 6 b 6 p − 1, 1 k k k ∂zbp = · ∂zp ◦ · · · ◦ ∂zp , (b times). b! Moreover the p-th power rule (see [3, §7], [15, §2.2]) says that for f ∈ F ((z − c)), (
ps
if j = ℓps , ∂zℓ (f ) j ps (2.14) ∂z f = 0 otherwise, and so calculation using (2.13) and (2.14) can often be fairly efficient. 3. Goss polynomials and hyperderivatives We review here results on Goss polynomials, which were introduced by Goss in [12, §6] and have been studied further by Gekeler [7, §3], [8]. We start first with an Fq -vector space Λ ⊆ C∞ of dimension d. We define the exponential function of Λ,  Y′  z , 1− eΛ (z) = z λ λ∈Λ

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

7

which is an Fq -linear polynomial of degree q d . If we take tΛ (z) = 1/eΛ (z), then just as in (2.11) we have X 1 1 = . tΛ (z) z−λ λ∈Λ

We can extend these definitions to any discrete lattice Λ ⊆ C∞ , which is the union of nested finite dimensional Fq -vector spaces Λ1 ⊆ Λ2 ⊆ · · · . We find that generally eΛ (z) = limi→∞ eΛi (z) and tΛ (z) = limi→∞ tΛi (z), where the convergence is coefficientwise in C∞ ((z)).

Remark 3.1. If we take Λ = ΛC , then eΛC (z) = eC (z), whereas if we take Λ = A, then eA (z) = πe1 eC (e π z). Thus π e tA (z) = π e tΛC (e π z) = , eC (e πz) and u(z), as defined in (2.6), is given by tA (z) u(z) = = tΛC (e π z). π e This normalization of u(z) is taken so that the u-expansions of some Drinfeld modular forms will have K-rational coefficients. Theorem 3.2 (Goss [12, §6]; see also Gekeler [7, §3]). Let Λ ⊆ C∞ be a discrete Fq -vector space. Let  X ∞ Y′  z j = αj z q , 1− eΛ (z) = z λ j=0 λ∈Λ

and let tΛ (z) = 1/eΛ (z). For each k > 1, there is a monic polynomial Gk,Λ (t) of degree k with coefficients in Fq [α0 , α1 , . . .] so that X
1 = G t (z) . Sk,Λ (z) := k,Λ Λ k (z − λ) λ∈Λ Furthermore the following properties hold. (a) Gk,Λ (t) = t(Gk−1,Λ (t) + α1 Gk−q,Λ (t) + α2 Gk−q2 ,Λ (t) + · · · ). (b) We have a generating series identity ∞ X tx GΛ (t, x) = Gk,Λ (t)xk = . 1 − teΛ (x) k=1 (c) If k 6 q, then Gk,Λ (t) = tk . (d) Gpk,Λ(t) = G
k,Λ (t)p . (e) t2 ∂t1 Gk,Λ (t) = kGk+1,Λ (t).

Gekeler [7, (3.8)] finds a formula for each Gk,Λ (t), k X  X j i j+1 (3.3) Gk+1,Λ (t) = αt , i j=0 i

where the sum is over all (s + 1)-tuples i = (i0 , .
. . , is ), with s arbitrary, satisfying i0 + · · · + is = j and i0 + i1 q + · · · + is q s = k; ji = j!/(i0 ! · · · is !) is a multinomial coefficient; and αi = α0i0 · · · αsis .

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MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

Part (e) of Theorem 3.2 indicates that there are interesting hyperderivative relations among Goss polynomials, with respect to t and to z, which we now investigate. All hyperderivatives we will take will be of polynomials and formal power series, but the considerations in §2 about holomorphic functions will play out later in the paper. The main result of this section is the following. Theorem 3.4. Let Λ ⊆ C∞ be a discrete Fq -vector space, and let t = tΛ (z). For r > 0, we define βr,j so that r X Gr+1,Λ (t) = βr,j tj+1 . j=0

Then

(3.4a) (3.4b)


∂zr (tn ) = (−1)r · tn ∂tn−1 tn−2 Gr+1,Λ (t) , ∂zr (tn ) = (−1)r

r X

βr,j tj+1 ∂tj (tn+j−1),

∀ n > 1, ∀ n > 0,

j=0

and (3.4c)

  r X
n+r−1 Gn+r,Λ (t) = βr,j tj+1 ∂tj tj−1 Gn,Λ (t) , r j=0

∀ n > 1.

Remark 3.5. We see that (3.4a) and (3.4b) generalize (2.12) and that (3.4c) generalizes Theorem 3.2(e). In later sections (3.4a) and (3.4b) will be useful for taking derivatives of Drinfeld modular forms. The coefficients βr,j can be computed using the generating series GΛ (t, x) or equivalently (3.3). The proof requires some preliminary lemmas. Lemma 3.6 (cf. Petrov [20, §3]). For r > 0 and n > 1,  
r r n+r−1 Gn+r,Λ (t). (3.6a) ∂z Sn,Λ (z) = (−1) r Moreover, we have

(3.6b) ∂zr Sn,Λ (z) = (−1)n+r−1 · ∂zn−1 Sr+1,Λ (z) and

(3.6c)

∂zr (t) = (−1)r Gr+1,Λ (t).

Proof. Using the power and quotient rules [15, §2.2], we see that for λ ∈ C∞ ,       1 1 1 −n r r n+r−1 ∂z = = (−1) . n n+r (z − λ) r (z − λ) r (z − λ)n+r Therefore,  
r r n+r−1 Sn+r,Λ (z), ∂z Sn,Λ (z) = (−1) r and combining with the defining property of Gn+r,Λ (t) in Theorem 3.2, we see that (3.6a) follows. Now     (r + 1) + (n − 1) − 1 n+r−1 , = n−1 r and so (3.6b) follows from (3.6a). Finally, (3.6c) is a special case of (3.6a) with n = 1. 

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

9

Lemma 3.7. For n > 1, we have an identity of rational functions in x,   n−1
t x x n−1 n−2 n−1 = ∂ t G (t, x) . = ∂ Λ t t (1 − teΛ (x))n 1 − teΛ (x)

Proof. Our derivatives with respect to t are taken while considering x to be a constant. We note that for ℓ > 0,   1 eΛ (x)ℓ ℓ ∂t = , 1 − teΛ (x) (1 − teΛ (x))ℓ+1 by the quotient and chain rules [15, §2.2]. Therefore, by the product rule,    X  n−1 tn−1 1 n−1 k n−1 n−1−k ∂t = ∂t (t )∂t 1 − teΛ (x) 1 − teΛ (x) k=0 n−1−k  n−1  X 1 teΛ (x) n−1 · = 1 − teΛ (x) 1 − teΛ (x) k k=0  n−1 teΛ (x) 1 = 1+ · . 1 − teΛ (x) 1 − teΛ (x)

A simple calculation yields that this is 1/(1 − teΛ (x))n , and the result follows.



Proof of Theorem 3.4. The chain rule [15, §2.2] and (3.6c) imply that r   X n n−k X r n ∂zℓ1 (t) · · · ∂zℓk (t) t ∂z (t ) = k ℓ ,...,ℓ >1 k=1 1

k

ℓ1 +···+ℓk =r

= (−1)

r

r   X n k=1

k

tn−k

X

Gℓ1 +1,Λ (t) · · · Gℓk +1,Λ (t).

ℓ1 ,...,ℓk >1 ℓ1 +···+ℓk =r

By direct expansion (see [15, §2.2, Eq. (I)]), the final inner sum above is the coefficient of xr in
k G2,Λ (t)x + G3,Λ (t)x2 + · · · , and therefore by the binomial theorem, 
n  r n r r 2 . ∂z (t ) = (−1) · coefficient of x in t + G2,Λ (t)x + G3,Λ (t)x + · · · Now G1,Λ (t) = t, so

2

t + G2,Λ (t)x + G3,Λ (t)x + · · · =

∞ X

Gk,Λ (t)xk−1 =

k=1

Therefore, ∂zr (tn )

 = (−1) · coefficient of xr+1 in r

From Lemma 3.7 we see that

GΛ (t, x) t = . x 1 − teΛ (x)

 tn x . (1 − teΛ (x))n

∞ X
k tn x n−1 n−2 n ∂ t G (t) x , = t k,Λ t (1 − teΛ (x))n k=1

10

MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

and so (3.4a) holds. To prove (3.4b), we first note that it holds when n = 0 by checking the various cases and using that βr,0 = 0 for r > 1, since Gr+1 (t) is divisible by t2 for r > 1 by Theorem 3.2, and that β0,0 = 1. For n > 1, we use (3.4a) and write  X r
r n r n n−1 n−2 r n n−1 n+j−1 ∂z (t ) = (−1) · t ∂t . t Gr+1,Λ (t) = (−1) · t ∂t βr,j t j=0

Noting that

∂tn−1 (tn+j−1 ) we then have ∂zr (tn )

  n+j−1 j t = tj−n+1 ∂tj (tn+j−1 ), = n−1 = (−1)

r

r X

βr,j tj+1 ∂tj (tn+j−1 ),

j=0

and so (3.4b) holds. Furthermore, by (3.6a) and (3.6b),  

n+r−1 Gn+r,Λ (t) = (−1)n−1 · ∂zn−1 Sr+1,Λ (z) = (−1)n−1 · ∂zn−1 Gr+1,Λ (t) . r But then by (3.4a), ∂zn−1




Gr+1,Λ (t) =

r X

βr,j ∂zn−1 (tj+1 )

= (−1)

n−1

j=0

r X j=0


βr,j tj+1 ∂tj tj−1 Gn,Λ (t) ,

which yields (3.4c).



4. Theta operators on Drinfeld modular forms We recall the definition of Drinfeld modular forms for GL2 (A), which were initially studied by Goss [10], [11], [12]. We will also review results on u-expansions of modular forms due to Gekeler [7]. Throughout we let Γ = GL2 (A). A holomorphic function f : Ω → C∞ is a Drinfeld modular form of weight k > 0 and type m ∈ Z/(q − 1)Z if (1) for all γ = ( ac db ) ∈ Γ and all z ∈ Ω, f (γz) = (det γ)−m (cz + d)k f (z),

γz =

az + b ; cz + d

(2) and f is holomorphic at ∞, i.e., f has a u-expansion and so f ∈ U(Ω). We let Mkm be the C∞ -vector space of modular forms of weight k and type m. We know L L ′ m+m′ that Mkm · Mkm′ ⊆ Mk+k and that M = k,m Mkm and M 0 = k Mk0 are graded C∞ ′ algebras. Moreover, in order to have Mkm 6= 0, we must have k ≡ 2m (mod q − 1). If L is a subring of C∞ , then we let Mkm (L) denote the space of forms with P u-expansion m m coefficients in L, i.e., Mk (L) = Mk ∩ L[[u]]. We note that if f = cn un is the uexpansion of f ∈ Mkm , then (4.1)

cn 6= 0

which can be seen by using γ =

⇒
ζ 0 0 1

n≡m

(mod q − 1),

, for ζ a generator of F× q , in the definition above.

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

11

Certain Drinfeld modular forms can be expressed in terms of A-expansions, which we now recall. For k > 1, we set (4.2)

Gk (t) = Gk,ΛC (t) =

k−1 X

βk−1,j tj+1 ,

j=0

to be the Goss polynomials with respect to the lattice ΛC . Since eC (z) ∈ K[[z]], it follows from Theorem 3.2 that the coefficients βk−1,j ∈ K for all k, j. As in (2.6) and Remark 3.1, we have u(z) = 1/eC (e π z), and for a ∈ A we set (4.3)

ua (z) := u(az) =

1 . eC (e π az)

Since eC (e π az) = Ca (eC (e π z)), if we take the reciprocal polynomial for Ca (z) to be Ra (z) = q deg a z Ca (1/z) then deg a

(4.4)

uq deg a ua = = uq + · · · ∈ A[[u]]. Ra (u)

We say that a modular form f has an A-expansion if there exist k > 1 and c0 , ca ∈ C∞ for a ∈ A+ , so that X ca Gk (ua ). f = c0 + a∈A+

Example 4.5. For k ≡ 0 (mod q − 1), k > 0, the primary examples of Drinfeld modular forms with A-expansions come from Eisenstein series, 1 1 X′ , Ek (z) = k π e a,b∈A (az + b)k

which is a modular form of weight k and type 0. Gekeler [7, (6.3)] showed that X X 1 X′ 1 ζC (k) (4.6) Ek = k G (u ) = − Gk (ua ), − − k a π e b∈A bk a∈A π ek a∈A+ + P −k where ζC (k) = is a Carlitz zeta value. We know (see [13, §9.2]) that a∈A+ a k ζC (k)/e π ∈ K. For more information and examples on A-expansions the reader is directed to Gekeler [7], L´opez [16], [17], and Petrov [19], [20]. Example 4.7. We can also define the false Eisenstein series E(z) of Gekeler [7, §8] to be 1 XX a , E(z) := π e a∈A b∈A az + b +

which is not quite a modular form but is a quasi-modular form similar to the classical weight 2 Eisenstein series [1], [7]. Gekeler showed that E ∈ U(Ω) and that E has an A-expansion, X X aua . aG1 (ua ) = (4.8) E= a∈A+

a∈A+

12

MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

We now define theta operators Θr on functions in H(Ω) by setting for r > 0, Θr :=

(4.9)

1 ∂zr . r (−e π)

If we take Θ = Θ1 , then by (2.12), Θu = u2 , and Θ plays the role of the classical theta d operator ϑ = q dq . Just as in the classical case, Θ and more generally Θr do not take modular forms to modular forms. However, Bosser and Pellarin [1, Thm. 2] prove that Θr preserves quasi-modularity: Θr : C∞ [E, g, h] → C∞ [E, g, h], where E is the false Eisenstein series, g = Eq−1 , and h is the cusp form of weight q + 1 and type 1 defined by Gekeler [7, Thm. 5.13] as the (q − 1)-st root of the discriminant function ∆. To prove their theorem, Bosser and Pellarin [1, Lem. 3.5] give formulas for Θr (un ), which are ostensibly a bit complicated. From Theorem 3.4, we have the following corollary, which perhaps conceptually simplifies matters. Corollary 4.10. For r > 0, (4.10a) (4.10b)


Θr (un ) = un ∂un−1 un−2 Gr+1 (u) , ∀ n > 1,  r  r X X n+j−1 r n j+1 j n+j−1 βr,j un+j , βr,j u ∂u (u )= Θ (u ) = j j=0 j=0

∀ n > 0,

where βr,j are the coefficients of Gr+1 (t) in (4.2).

Proof. The proof of (4.10a) is straightforward, but it is worth noting how the different normalizations of u(z) and tΛC (z) work out. From Remark 3.1, we see that  r  r
−1 −1 r n r n ∂z tΛC (e π z) = ·π er ∂zr tΛC ) z=eπz Θ (u ) = π e π e

n n−1 n−2 = t ∂t t Gr+1 (t) t=t (eπ z) = un ∂un−1 un−2 Gr+1 (u) , ΛC

where the third equality is (3.4a). The proof of (4.10b) is then the same as for (3.4b).  Remark 4.11. We see from (4.10a) that there is a duality of some fashion between the r-th derivative of un and the (n − 1)-st derivative of Gr+1 (u), which dovetails with (3.6b). We see from this corollary that Θr can be seen as the operator on power series in C∞ [[u]] given by the following result. Moreover, from (4.12b), we see that computation of Θr (f ) is reasonably straightforward once the computation of the coefficients of Gr+1 (t) can be made. P Corollary 4.12. Let f = cn un ∈ U(Ω). For r > 0, (4.12a)

r

r

Θ (f ) = Θ (c0 ) +

∞ X n=1

(4.12b)

Θr (f ) =

r X j=0


cn un ∂un−1 un−2Gr+1 (u) ,


βr,j uj+1 ∂uj uj−1f ,

where βr,j are the coefficients of Gr+1 (t) in (4.2).

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

13

Finally we recall the definition of the r-th Serre operator D r on modular forms in Mkm for r > 0. We set   r X r r i k+r−1 Θr−i (f )Θi−1 (E). (4.13) D (f ) := Θ (f ) + (−1) i i=1 The following result shows that D r takes modular forms to modular forms.

Theorem 4.14 (Bosser-Pellarin [2, Thm. 4.1]). For any weight k, type m, and r > 0,
m+r D r Mkm ⊆ Mk+2r . 5. v-adic modular forms

In this section we review the theory of v-adic modular forms introduced by Goss [14] and Vincent [24]. In [21], Serre defined p-adic modular forms as p-adic limits of Fourier series of classical modular forms and determined their properties, in particular their behavior under the ϑ-operator. For a fixed finite place v of K, Goss and Vincent recently transferred Serre’s definition to the function field setting of v-adic modular forms, and Goss produced families of examples based on work of Petrov [19] (see Theorem 6.3). In §6, we show that v-adic modular forms are invariant under the operators Θr . For our place v of K we fix ℘ ∈ A+ , which is the monic irreducible generator of the ideal pv associated to v, and we let d := deg(℘). As before we let Av and Kv denote completions with respect to v. We will write K⊗Av [[u]] for K⊗A Av [[u]], and we recall that K⊗Av [[u]] be identified Pcan ∞ with elements of Kv [[u]] that have bounded denominators. For f = n=0 cn un ∈ K ⊗ Av [[u]], we set (5.1)

ordv (f ) := inf {ordv (cn )} = min{ordv (cn )}. n

n

If ordv (f ) > 0, i.e., if f ∈ Av [[u]], then we say f is v-integral. For f , g ∈ K ⊗ Av [[u]], we write that f ≡ g (mod ℘m ), if ordv (f − g) > m. We also define a topology on K ⊗ Av [[u]] in terms of the v-adic norm, kf kv := q − ordv (f ) ,

(5.2)

which is a multiplicative norm by Gauss’ lemma. Following Goss, we define the v-adic weight space S = Sv by (5.3)

S := lim Z/(q d − 1)pℓ Z = Z/(q d − 1)Z × Zp . ←− ℓ

We have a canonical embedding of Z ֒→ S, by identifying n ∈ Z with (n, n), where n is the class of n modulo q d − 1. For any a ∈ A+ with ℘ ∤ a, we can decompose a as a = a1 a2 , d × where a1 ∈ A× v is the (q − 1)-st root of unity satisfying a1 ≡ a (mod v) and a2 ∈ Av satisfies a2 ≡ 1 (mod v). Then for any s = (x, y) ∈ S, we define (5.4)

as := ax1 ay2 .

This definition of as is compatible with the usual definition when s is an integer. Furthermore, it is easy to check that the function s 7→ as is continuous from S to A× v.

14

MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

Definition 5.5 (Goss [14, Def. 5]). We say a power series f ∈ K ⊗ Av [[u]] is a v-adic modular form of weight s ∈ S, in the sense of Serre, if there exists a sequence of K-rational modular forms fi ∈ Mkmi i (K) so that as i → ∞, (a) kfi − f kv → 0, (b) ki → s in S. Moreover, if f 6= 0, then mi is eventually a constant m ∈ Z/(q − 1)Z, and we say that m is the type of f . We say that fi converges to f as v-adic modular forms. It is easy to see that the sum and difference of two v-adic modular forms, both with weight s and type m, are also v-adic modular forms with the same weight and type. We set  (5.6) Mm s = f ∈ K ⊗ Av [[u]] f a v-adic modular form of weight s and type m , which is a Kv -vector space, and we note that

m2 m1 +m2 1 Mm s1 · Ms2 ⊆ Ms1 +s2 . m We take Mm s (Av ) := Ms ∩ Av [[u]], which is an Av -module. Moreover, any Drinfeld m modular form in Mk (K) is also a v-adic modular form as the limit of the constant sequence (u-expansion coefficients of forms in Mkm (K) have bounded denominators by [7, Thm. 5.13, §12], [11, Thm. 2.23]), and so for k ∈ Z, k > 0,

Mkm (K) ⊆ Mm k ,

Mkm (A) ⊆ Mm k (Av ).

The justification of the final part of Definition 5.5 is the following lemma. Lemma 5.7. Suppose that fi ∈ Mkmi i (K) converge to a non-zero v-adic modular form f . Then there is some m ∈ Z/(q − 1)Z so that except for finitely terms mi = m. P Proof. Since kf − fi kv → 0, it follows that kfi − fj kv → 0 as i, j → ∞. If f = cn u n and cn 6= 0, then from (4.1) we see that for i, j ≫ 0, n ≡ mi ≡ mj (mod q − 1).  Proposition 5.8. Suppose {fi } is a sequence of v-adic modular forms with weights si . Suppose that we have f0 ∈ K ⊗ Av [[u]] and s0 ∈ S satisfying, (a) kfi − f0 kv → 0, (b) si → s0 in S. Then f0 is a v-adic modular form of weight s0 . The type of f0 is the eventual constant type of the sequence {fi }. Proof. For each i > 1, we have a sequence of Drinfeld modular forms gi,j → fi as j → ∞. Standard arguments show that the sequence of Drinfeld modular forms {gi,i }∞ i=1 converges to f0 with respect to the k · kv -norm and that the weights ki of gi,i go to s0 in S.  We recall the definitions of Hecke operators on Drinfeld modular forms and their actions on u-expansions [7, §7], [12, §7]. For ℓ ∈ A+ irreducible of degree e, the Hecke operator Tℓ : Mkm → Mkm is defined by X z + β  k k f (Tℓ f )(z) = ℓ f (ℓz) + Uℓ f (z) = ℓ f (ℓz) + . ℓ β∈A ( 1, we let (5.13)

gd (z) = −Ld · Eqd −1 (z),

which is a Drinfeld modular form of weight q d −1 and type 0. By the following proposition we see that gd plays the role of Ep−1 for classical modular forms. Proposition 5.14 (Gekeler [7, Prop. 6.9, Cor. 6.12]). For d > 1, the following hold: (a) gd ∈ A[[u]]; (b) gd ≡ 1 (mod [d]). Proof of Proposition 5.12. Let f ∈ Mm s . Once we establish that Tℓ (f ), U℘ (f ), and V℘ (f ) , we claim the statement about the operators preserving Mm are elements of Mm s (Av ) is s a consequence of (5.9)–(5.11). Indeed, in either case ℓ 6= ℘ or ℓ = ℘ we have Vℓ (Av [[u]]) ⊆ Av [[u]], since in (5.10) the unℓ terms are in A[[u]] by (4.4). Likewise for Uℓ , the polynomials Gn,Λℓ (u) in (5.9) are in A[u], as the finite dimensinoal Fq -lattice Λℓ has exponential function given by polynomials from the Carlitz action, eΛℓ (z) = Cℓ (z) ∈ A[z], and thus by Theorem 3.2 we see that Gn,Λℓ (u) ∈ A[u]. Additionally we recall that the cases of U℘ and V℘ were previously proved by Vincent [24, Cor. 3.2, Prop. 3.3]. Now by hypothesis we can choose a sequence {fi } of Drinfeld modular forms of weight ki and type m so that fi → f and ki → s. By Proposition 5.14(b), for any i > 0, i

i

gdq ≡ 1 (mod ℘q ), i

i

since ordv ([d]) = 1. The form gdq has weight (q d −1)q i and type 0, and certainly fi gdq → f with respect to the k · kv -norm. However, we also have that as real numbers, i

weight of fi gdq = ki + (q d − 1)q i → ∞,

as i → ∞.

16

MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

Therefore, it suffices P to assume that Pki → n∞ as real numbers, as i → ∞. Suppose that f = cn un , fi = cn,i u ∈ K ⊗ Av [[u]]. For ℓ 6= ℘, since ℓki → ℓs and cn,i → cn , we have ∞ ∞ X X an,i unℓ + cn,i Gn,Λℓ (ℓu) −→ Tℓ (f ). Tℓ (fi ) = ℓki n=0

n=0

Mkmi (K),

Since Tℓ (fi ) ∈ it follows that Tℓ (f ) ∈ Mm s . Now consider the case ℓ = ℘. Since ki → ∞, we see that |℘ki |v → 0. Therefore, ∞ X T℘ (fi ) → cn Gn,Λ℘ (℘u) = U℘ (f ), n=0

Mm s .

and so U℘ (f ) ∈ By the same argument each U℘ (fi ) ∈ Mm ki , starting with the constant sequence fi in the first paragraph. By subtraction each
(5.15) V℘ (fi ) = ℘−ki T℘ (fi ) − U℘ (fi )

is then an element of Mm ki . Because cn,i → cn , we see from (5.10) that V℘ (fi ) → V℘ (f ) with respect to the k · kv -norm. Thus by Proposition 5.8, V℘ (f ) ∈ Mm  s as desired. 6. Theta operators on v-adic modular forms As is well known the operators Θr do not generally take Drinfeld modular forms to Drinfeld modular forms [1], [22]. However, we will prove in this section that each Θr , r > 0, does preserve spaces of v-adic modular forms. Using the equivalent formulations in (4.12a) and (4.12b), we define Kv -linear operators Θr : K ⊗ Av [[u]] → K ⊗ Av [[u]],

r > 0.

Theorem 6.1. For any weight s ∈ S and type m ∈ Z/(q − 1)Z, we have for r > 0, m+r Θr : M m s → Ms+2r .

This can be seen as similar in spirit to the results of Bosser and Pellarin [1, Thm. 2], [2, Thm. 4.1] (see also Theorem 4.14), that Θr preserves spaces of Drinfeld quasi-modular forms, and our main arguments rely on essentially showing that quasi-modular forms with Kv -coefficients are v-adic and applying Theorem 4.14. Consider first the operator Θ = Θ1 , which can be equated by (2.12) with the operation on u-expansions given by Θ = u2 ∂u1 . We recall a formula of Gekeler [7, §8] (take r = 1 in (4.13)), which states that for f ∈ Mkm , Θ(f ) = D 1 (f ) + kEf, where E is the false Eisenstein series whose u-expansion is given in (4.8). Our first goal is to show that E is a v-adic modular form, for which we use results of Goss and Petrov. For k, n > 1 and s ∈ S, we set X X as Gn (ua ). ak−n Gn (ua ), fˆs,n := (6.2) fk,n := a∈A+

a∈A+ ℘∤a

The notation fk,n and fˆs,n is not completely consistent, since fk,n is more closely related to fˆk−n,n than fˆk,n , but this viewpoint is convenient in many contexts (see [14]).

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

17

Theorem 6.3 (Goss [14, Thm. 2], Petrov [19, Thm. 1.3]). (a) (Petrov) Let k, n > 1 be chosen so that k − 2n > 0, k ≡ 2n (mod q − 1), and n 6 pordp (k−n) . Then fk,n ∈ Mkm (K), where m ≡ n (mod q − 1). (b) (Goss) Let n > 1. For s = (x, y) ∈ S with x ≡ n (mod q − 1) and y ≡ 0 (mod q ⌈logq (n)⌉ ), we have fˆs,n ∈ Mm s+n , where m ≡ n (mod q − 1). We note that the statement of Theorem 6.3(b) is slightly stronger than what is stated in [14], but Goss’ proof works here without changes. We then have the following corollary. Corollary 6.4. For any ℓ ≡ 0 (mod q − 1), we have fˆℓ+1,1 ∈ M1ℓ+2 . If we take ℓ = 0, we see that fˆ1,1 =

X

aua ∈ Av [[u]]

a∈A+ ℘∤a

is a v-adic modular form in M12 (Av ) and is a partial sum of E in (4.8). From this we can prove that E itself is a v-adic modular form. Theorem 6.5. The false Eisenstein series E is a v-adic modular form in M12 (Av ). Proof. Starting with the expansion in (4.8), we see that E ∈ Av [[u]]. Also, X X X au℘a aua + ℘ aua = E= a∈A+

a∈A+ ℘∤a

a∈A+

=

X

aua + ℘

X

au℘a + ℘2

a∈A+ ℘∤a

a∈A+ ℘∤a

E=

℘j

j=0

We note that

X

au℘j a = V℘◦j

X

a∈A+ ℘∤a

a∈A+ ℘∤a

!

au℘j a .

a∈A+ ℘∤a

X

au℘2 a ,

a∈A+

and continuing in this way, we find ∞ X

X

aua

!

= V℘◦j (fˆ1,1 ),

where V℘◦j is the j-th iterate V℘ ◦ · · · ◦ V℘ . By Proposition 5.12, we see that V℘◦j (fˆ1,1 ) ∈ M12 (Av ) for all j. Moreover, ∞ X E= ℘j V℘◦j (fˆ1,1 ), j=0

the right-hand side of which converges with respect to the k · kv -norm, and so we are done by Proposition 5.8. 

18

MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

m Proof of Theorem 6.1. Let f ∈ Mm s and pick fi ∈ Mki (K) with fi → f . It follows from r r the formulas in Corollary 4.12 that Θ (fi ) → Θ (f ) with respect to the k · kv -norm for each r > 0, so by Proposition 5.8 it remains to show that each

Θr (fi ) ∈ Mm+r ki +2r . We proceed by induction on r. If r = 1, then since D 1 (fi ) ∈ Mkm+1 (K) for each i by i +2 Theorem 4.14, it follows from Theorem 6.5 that Θ(fi ) = D 1 (fi ) + ki Efi ∈ Mm+1 ki +2 , for each i. Now by (4.13), for each i   r X j ki + r − 1 r r Θr−j (fi )Θj−1(E). Θ (fi ) = D (fi ) − (−1) j j=1

By Theorem 4.14, D r (fi ) ∈ Mkm+r (K), and by the induction hypothesis and Theorem 6.5 i +2r the terms in the sum are in Mm+r  ki +2r . 7. Theta operators and v-adic integrality m+r We see from Theorem 6.1 that Θr : Mm s → Ms+2r , and it is a natural question to ask r whether Θ preserves v-integrality, i.e., ?

m+r Θr : M m s (Av ) → Ms+2r (Av ).

However, it is known that this can fail for r sufficiently large because of the denominators in Gr+1 (u) (e.g., see Vincent [25, Cor. 1]). Nevertheless, in this section we see that Θr is not far off from preserving v-integrality. For an A-algebra R and a sequence {bm } ⊆ R, we define an A-Hurwitz series over R (cf. [13, §9.1]) by ∞ X bm m (7.1) h(x) = x ∈ (K ⊗A R)[[x]], Πm m=0

where we recall the definition of the Carlitz factorial Πm from (2.3). Series of this type were initially studied by Carlitz [4, §3] and further investigated by Goss [9, §3], [13, §9.1]. The particular cases we are interested in are when R = A or R = A[u], but we have the following general proposition whose proof can be easily adapted from [9, §3.2], [13, Prop. 9.1.5]. Proposition 7.2. Let R be an A-algebra, and let h(x) be an A-Hurwitz series over R. (a) If the constant term of h(x) is 1, then 1/h(x) is also an A-Hurwitz series over R. (b) If g(x) is an A-Hurwitz series over R with constant term 0, then h(g(x)) is also an A-Hurwitz series over R. We apply this proposition to the generating function of Goss polynomials. Lemma 7.3. For each k > 1, we have Πk−1 Gk (u) ∈ A[u]. Proof. Consider the generating series ∞ G(u, x) X u = Gk (u)xk−1 = . x 1 − ueC (x) k=1

THETA OPERATORS, GOSS POLYNOMIALS, AND v-ADIC MODULAR FORMS

19

We claim that G(u, x)/x is an A-Hurwitz series over A[u]. Indeed certainly the constant series u itself is one, and i ∞ X uxq 1 − ueC (x) = 1 − Πqi i=0

is an A-Hurwitz series over A[u] with constant term 1, so the claim follows from Proposition 7.2(a). The result is then immediate.  m+r r Theorem 7.4. For r > 0, if f ∈ Mm s (Av ), then Πr Θ (f ) ∈ Ms+2r (Av ). Thus we have a well-defined operator, m+r Πr Θr : Mm s (Av ) → Ms+2r (Av ).

Proof. By (4.12a), we see that the possible denominators of Θr (f ) come from the denominators of Gr+1 (u), which are cleared by Πr using Lemma 7.3.  Remark 7.5. Once we see that Πr Θr preserves v-integrality, the question of whether Πr is the best possible denominator is important but subtle, and in general the answer is no. For example, taking r = q d+1 − 1, we see from Theorem 3.2(d), that Gqd+1 (u) = uq

d+1

,

q d+1 −1

and so Θ : Av [[u]] → Av [[u]] already by (4.12a). However, Πqd+1 −1 can be seen to be divisible by ℘. Nevertheless, we do see that Πr is the best possible denominator in many cases. For example, let r = q i for i > 1. Then from Theorem 3.2(a),     Gqi +1−q (u) Gqi +1−q (u) G1 (u) u qi Gqi +1 (u) = u Gqi (u) + +···+ =u u + + ···+ . D1 Di D1 Di

From Theorem 3.2(b) we know that u2 divides Gk (u) for all k > 2, and so we find that the coefficient of u2 in Gqi +1 (u) is precisely 1/Di , which is the same as Πqi . References

[1] V. Bosser and F. Pellarin, Hyperdifferential properties of Drinfeld quasi-modular forms, Int. Math. Res. Not. IMRN 2008 (2008), Art. ID rnn032, 56 pp. [2] V. Bosser and F. Pellarin, On certain families of Drinfeld quasi-modular forms, J. Number Theory 129 (2009), no. 12, 2952–2990. [3] W. D. Brownawell, Linear independence and divided derivatives of a Drinfeld module. I. in: Number Theory in Progress, Vol. 1 (Zakopane-Ko´scielisko, 1997), de Gruyter, Berlin, 1999, pp. 47–61. [4] L. Carlitz, An analogue of the von Staudt-Clausen theorem, Duke Math. J. 3 (1937), no. 3, 503–517. [5] K. Conrad, The digit principle, J. Number Theory 84 (2000), no. 2, 230–257. [6] J. Fresnel and M. van der Put, Rigid Analytic Geometry and its Applications, Birkh¨auser, Boston, 2004. [7] E.-U. Gekeler, On the coefficients of Drinfeld modular forms, Invent. Math. 93 (1988), no. 3, 667– 700. [8] E.-U. Gekeler, On the zeroes of Goss polynomials, Trans. Amer. Math. Soc. 365 (2013), no. 3, 1669–1685. [9] D. Goss, von Staudt for Fq [T ], Duke Math. J. 45 (1978), no. 4, 885–910. [10] D. Goss, Modular forms for Fr [T ], J. reine Angew. Math. 317 (1980), 16–39. [11] D. Goss, π-adic Eisenstein series for function fields, Compos. Math. 41 (1980), no. 1, 3–38. [12] D. Goss, The algebraist’s upper half-plane, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 3, 391–415. [13] D. Goss, Basic Structures of Function Field Arithmetic, Springer-Verlag, Berlin, 1996. [14] D. Goss, A construction of v-adic modular forms, J. Number Theory 136 (2014), 330–338.

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MATTHEW A. PAPANIKOLAS AND GUCHAO ZENG

[15] S. Jeong, Calculus in positive characteristic p, J. Number Theory 131 (2011), no. 6, 1089–1104. [16] B. L´ opez, A non-standard Fourier expansion for the Drinfeld discriminant function, Arch. Math. (Basel) 95 (2010), no. 2, 143–150. [17] B. L´ opez, Action of Hecke operators on two distinguished Drinfeld modular forms, Arch. Math. (Basel) 97 (2011), no. 5, 423–429. [18] M. A. Papanikolas, Log-algebraicity on tensor powers of the Carlitz module and special values of Goss L-functions, in preparation. [19] A. Petrov, A-expansions of Drinfeld modular forms, J. Number Theory 133 (2013), no. 7, 2247– 2266. [20] A. Petrov, On hyperderivatives of single-cuspidal Drinfeld modular forms with A-expansions, J. Number Theory 149 (2015), 153–165. [21] J.-P. Serre, Formes modulaires et fonctions zˆeta p-adiques, in: Modular functions of one variable, III (Proc. Internat. Summer School, Univ. Antwerp, 1972), Springer, Berlin, 1973, pp. 191–268. [22] Y. Uchino and T. Satoh, Function field modular forms and higher-derivations, Math. Ann. 311 (1998), no. 3, 439–466. [23] C. Vincent, Drinfeld modular forms modulo p, Proc. Amer. Math. Soc. 138 (2010), no. 12, 4217– 4229. [24] C. Vincent, On the trace and norm maps from Γ0 (p) to GL2 (A), J. Number Theory 142 (2014), 18–43. [25] C. Vincent, Weierstrass points on the Drinfeld modular curve X0 (p), Res. Math. Sci. 2 (2015), 2:10, 40 pp. Department of Mathematics, Texas A&M University, College Station, TX 77843, USA E-mail address: [email protected] Department of Mathematics, Texas A&M University, College Station, TX 77843, USA E-mail address: [email protected]